Suppose f: D --> R is a function whose domain contains [a,b] for some a,b R amd f is integrable on [a,b]. suppose a < c < b.

1. Suppose A is a set of real number that is bounded above. Prove that for any > 0 there is a number a A Such that supA-a < .

(Hint: Prove by contradiction. If there is an >0 such that supA-a >= for all a A then what does this say about the number supA - ?)